The Owls won it thanks to a goalkeeping howler from the Tiger's Eldin Jakupovic, who inexplicably dropped Miguel Llera's corner over the line and then had to leave the field wearing a neck brace after Reda Johnson had clattered into him.
Hull wanted a foul for Johnson's challenge, which led to a 10-minute stoppage, but Haywood waved away appeals and Wednesday made sure of the points in the seventh minute of injury time when Michail Antonio raced clear and rolled home.
Wednesday's second goal had come just four minutes after Hull had drawn level thanks to Robert Koren's header, which cancelled out Johnson's first-half opener for Wednesday.
That was not Haywood's only involvement in the game, though, as the West Yorkshire official inexplicably ruled out a Giles Coke effort for an apparent foul on the uneasy Jakupovic.
Whatever the circumstances of Wednesday's win, it was thoroughly deserved as despite only playing winger Antonio in a lone striker's role, they put in a superb display and could have won even more convincingly.
The Owls settled well and were only denied an early lead by Jakupovic's fingertips after he superbly kept out Jermaine Johnson's fizzing 20-yard effort.
Wednesday forged ahead in the 23rd minute when an unmarked Johnson glanced in Rhys McCabe's free-kick through the legs of Jakupovic, who should have done better.
After a brief spell of Hull pressure at the start of the second period, Wednesday were denied a second in controversial circumstances.
After some shoddy defending in the area, Coke nipped in ahead of Jakupovic and clipped the ball home. The majority of the Wednesday players were celebrating in a heap in front of their travelling support when referee Haywood ruled Coke had fouled the keeper.
The Tigers had not tasted defeat on home soil since late November and looked as if they would keep that unbeaten run in tact with a goal out of nothing when Ahmed Elmohamady whipped in a superb cross and Koren glanced a header past Kirkland.
But just as Hull looked like they might go on and win it, Wednesday restored their lead after Jakupovic dropped Llera's corner over the line and the win was made safe when Antonio broke and rounded substitute goalkeeper Mark Oxley to roll home.
The home side's misery was complete when Jay Simpson's penalty was saved by Chris Kirkland.